Identification and Estimation of Dynamic Causal Effects in...

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Identification and Estimation of Dynamic Causal Effects in Macroeconomics

Jim Stock and Mark Watson

IAAE 2017 Sapporo

June 26, 2017

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Causal effects and IV regression

Avg. Causal Effect: E(Y | X = 1) – E(Y | X = 0)

Linear Regression: Y = γ + β X + u E(u | X) = 0 With Controls: Y = β X + γ’W + u X⊥ = X – Proj(X|W) E(u⊥X⊥) = 0 With Instruments, Z:

Relevance: E(X⊥Z⊥) ≠ 0 Exogeneity: E(u⊥Z⊥ ) = 0

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Dynamic causal effects and IRFs

Yt = Θ0 εt + Θ1 εt −1 + … = Θ(L)εt nY nε εj,t are "structural shocks": i.i.d. (over j and t) with mean zero. Dynamic ACE:

E(Yi,t+h | εj,t = 1) − E(Yi,t+h | εj,t = 0) = Θh,ij

( = ∂Yi,t+h / ∂ε j ,t )

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Estimating dynamic causal effects in macroeconomics Standard Approach:

• Estimate VAR for Y • Assume "invertibility" to relate εt to VAR forecast

errors. • Impose some restrictions on Θ(L) for identification

Alternative Approach:

• Find an "external" instrument Z that captures exogenous variation in one of the structural shocks.

• Use instrument (with or without VAR step) to estimate dynamic causal effects.

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Some references on external instruments

VARs: Stock (2008), Stock and Watson (2012), Mertens and Ravn (2013, 2014), Gertler and Karadi (2015), Caldera and Kamps (2017), Montiel Olea, Stock and Watson (2012), Lumsford (2015), Jentsch and Lunsford (2016), Carriero, Momtaz, Theodoridis and Theophilopoulou (2015), … Local-projections: Jordà, Schularick, and Taylor (2015), Ramey and Zubairy (2017), Ramey (2016), Mertens (2015), Fieldhouse, Mertens, Ravn (2017) …

This talk: Survey-ish

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A Running Empirical Example: Gertler-Karadi (2015)

• Yt = (Rt, 100×Δln(IP), 100×Δln(CPI), EBP)

• Monetary policy shock = ε1,t

• Causal Effects: E(Yi,t+h | ε1,t = 1) − E(Yi,t+h | ε1,t = 0) = Θh,i

• Kuttner (2001)-like instrument, Zt = change in Federal

Funds rate futures in short window around FOMC announcements.

o Zt correlated with ε1,t but uncorrelated with

ε•,t = (ε2,t ,ε3,t ,…,εnε ,t ).

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ε1,t isDirect estimation of Θh,i1

Yt = Θ0 εt + Θ1 εt −1 + … = Θ(L)εt

Yi,t+h = Θh,i1 ε1,t + ut (LP)

ut = {εt+h, … , εt+1, ε•,t , εt −1, … }

E(ε1,tut) = 0

But ε1,t is not observed

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Unit-Effect Normalization – 1

GK – IRF or Monetary Policy Shock

VOL.7 NO.1 61GERTLER AND KARADI: MONETARY POLICY SURPRISES

We begin with the external instruments case. As noted earlier, we use the three month ahead funds rate future surprise FF4 to identify monetary policy shock. As a check to ensure that this instrument is valid, we report the F-statistic from the first stage regression of the one-year bond rate residual on FF4. We find an F-value of 21 and half. We also compute a robust F-statistic (which allows for heteroskedasticity) of 17.5. Both values are safely above the threshold suggested by Stock et al. (2002) to rule out a reasonable likelihood of a weak instruments problem.

As the top left panel shows, a one standard deviation surprise monetary tight-ening induces a roughly 25 basis point increase in the one-year government bond rate. Consistent with conventional theory, there is a significant decline in industrial production that reaches a trough roughly a year and a half after the shock. Similarly consistent with standard theory, there is a small decline in the consumer price index that is not statistically significant. Note that in contrast to the Cholesky identifica-tion, we do not impose zero restrictions on the contemporaneous responses of output and inflation. The identification of the monetary policy shock is entirely due to the external instrument.

regression is incorporated in the reported confidence bands, because both stages of the estimation are included in the bootstrapping procedure. Thereby, we avoid any potential “generated regressor” problem.

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First-stage regression: F: 21.55; Robust F: 17.64; R

2: 7.76 percent; Adjusted R 2: 7.40 percent

External instruments Cholesky

Figure 1. One-Year Rate Shock with Excess Bond Premium





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Unit-Effect Normalization - 1

Parameters of interest: ∂ IPt+h / ∂ε1,t

∂Rt / ∂ε1,t





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Unit-effect normalization − 2

Θ0,11 = 1

GK: Y1,t = Rt and units are such that ∂Rt / ∂ε1,t = 1

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IV estimation of Θh,i2

Yi,t+h = Θh,i1ε1,t + {εt+h, … , εt+1, ε•,t , εt −1, … }

Y1,t = Θ0,11ε1,t + { ε•,t , εt −1, … } = ε1,t + { ε•,t , εt −1, … }

Yi,t+h = Θh,i1Y1,t + {εt+h, … , εt+1, ε•,t , εt −1, … }

Condition LP-IV: (i) E(ε1,t Zt) = α ≠ 0 (ii) E( ε•,t Zt') = 0 (iii) E(εt+j Zt') = 0 for j ≠ 0

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Odds and ends • HAR SEs

• Dyn. Causal Effects for levels vs. differences

• Weak-instrument robust inference

• "News" Shocks o replace Θ0,11 = 1 normalization with Θk,11 = 1

normalization

• Smoothness constraints (B&B − today, Plagborg-Møller,)

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Results for [R and 100 ×ln(IP)] (1990m1 -2012:m6)

lag (h) (a) R 0 1.00 (0.00) 6 -0.07 (1.34) 12 -1.05 (2.51) 24 -2.09 (5.66) IP 0 -0.59 (0.71) 6 -2.15 (3.42) 12 -3.60 (6.23) 24 -2.99 (10.21) Controls none First-stage F 1.7

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IV Estimation of Θh,i2 with additional controls -1

Yi,t+h = Θh,i1Y1,t + {εt+h, … , εt+1, ε•,t , εt −1, … }

2 Motivations:

(1) add controls to eliminate part of error term • controls should be uncorrelated with ε1,t. o Examples: lags of Z, Y, other macro variables

(2) Zt may be correlated with error, but uncorrelated after adding controls

(a) Example: GK Z = {ΔFFFt, ΔFFFt −1}. Add lags of FFFt.

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IV Estimation of Θh,i2 with additional controls - 2

Yi,t+h = Θh,i1Y1,t + γ'Wt + ut

xt⊥ = xt − Proj(xt | Wt)

Condition LP-IV⊥

(i) E ε1,t

⊥ Zt⊥′( ) = ′α ≠ 0

(ii) E ε•,t

⊥ Zt⊥′( ) = 0

(iii) E ε t+ j

⊥ Zt⊥′( ) = 0 for j ≠ 0.

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Results for [R and 100 ×ln(IP)] Yi,t+h = Θh,i1Y1,t + γ'Wt + {εt+h, … , εt+1, ε•,t , εt −1, … }

lag (h) (a) (b) (c) R 0 1.00 (0.00) 1.00 (0.00) 1.00 (0.00) 6 -0.07 (1.34) 1.12 (0.52) 0.67 (0.57) 12 -1.05 (2.51) 0.78 (1.02) -0.12 (1.07) 24 -2.09 (5.66) -0.80 (1.53) -1.57 (1.48) IP 0 -0.59 (0.71) 0.21 (0.40) 0.03 (0.55) 6 -2.15 (3.42) -3.80 (3.14) -4.05 (3.65) 12 -3.60 (6.23) -6.70 (4.70) -6.86 (5.49) 24 -2.99 (10.21) -9.51 (7.70) -8.13 (7.62) Controls none 4 lags of (z,y)

4 lags of (z,y,f)

First-stage F 1.7 23.7 18.6

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4 lags of (z,y,f)


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4 lags of (z,y,f)


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SVARs with External Instruments - 1

VAR: Yt = A1Yt-1 + A2Yt-2 + … + vt

Structural MA: Yt = Θ0 εt + Θ1 εt −1 + … = Θ(L)εt

Invertibility: εt = Proj(εt|Yt, Yt-1, … ) ⇒

vt = Θ0εt with Θ0 nonsingular (so ny = nε)

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SVARs with External Instruments - 2

A(L)Yt = vt = Θ0εt

⇒ Yt = C(L)Θ0εt with C(L)=A(L)-1

thus Θh,i1 = ChΘ0,i1

Unit-effect normalization yields: vi,t = Θ0,i1v1,t + { ε•,t }

Condition SVAR-IV (i) E(ε1,tZt) = α ≠ 0 (ii) E( ε•,t Zt') = 0

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SVAR with external instruments – estimation

1. Regress Yi,t onto Y1,t using instruments Zt and p lags of Yt as controls. This yields Θ0,i1

SVAR− IV . 2. Estimate a VAR(p) and invert the VAR to obtain C(L) = A(L)−1 . 3. Estimate the dynamic causal effects of shock 1 on the vector of variables as

Θh,1SVAR− IV = ChΘ0,1

SVAR− IV

(odds and ends: (1) News shocks; (2) Dif. sample periods in (1) and (2))

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SVAR with external instruments – inference

• Strong instruments:

T

A− AΘ0,1

SVAR− IV −Θ0,1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

d⎯ →⎯ Normal + δ-method

• Weak instruments: o T ( A− A) d⎯ →⎯ Normal. o Θ0,1

SVAR− IV −Θ0,1d⎯ →⎯ NonNormal.

o Use weak-instrument robust methods. (Montiel Olea, Stock and Watson (20xx)).

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SVAR with external instruments – invertibility 1

VAR forecast error: vt = Yt − Proj(Yt | Yt-1 ) Structural MA: Yt = Θ(L)εt

⇒

vt = Θ0εt +

Θh ε t−h − Proj(ε t−h |Y t−1)( )h=1∑

= Θ0εt + "omitted variables" Invertibility: εt = Proj(εt | Yt −1 ), so omitted variables vanish.

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SVAR with external instruments – invertibility 2 εt = Proj(εt | Yt −1 ) ⇒

• Proj(Yt| Yt-1, ε t−1) = Proj(Yt| Yt-1 )

• Proj(Yt| Yt-1, X t−1) = Proj(Yt| Yt-1 )

"Getting to invertibility"

• Many Ys • But depends on timing dynamic causal effects (Θ(L)) • No guarantees

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Results for [R and 100 ×ln(IP)]

lag (h) LP-IV 1990m1-2012m6

SVAR-IV IV: 1990m1-2012m6

VAR:1980m7-2012m6 R 0 1.00 (0.00) 1.00 (0.00) 6 1.12 (0.52) 0.89 (0.31) 12 0.78 (1.02) 0.78 (0.46) 24 -0.80 (1.53) 0.40 (0.49) IP 0 0.21 (0.40) 0.16 (0.59) 6 -3.80 (3.14) -0.81 (1.19) 12 -6.70 (4.70) -1.87 (1.54) 24 -9.51 (7.70) -2.16 (1.65) Controls 4 lags of (Z,Y)

12 lags of Y 4 lags of Z

First-stage F 23.7 20.5

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lag (h) LP-IV 1990m1-2012m6

SVAR-IV IV: 1990m1-2012m6

VAR:1980m7-2012m6 R 0 1.00 (0.00) 1.00 (0.00) 6 1.12 (0.52) 0.89 (0.31) 12 0.78 (1.02) 0.78 (0.46) 24 -0.80 (1.53) 0.40 (0.49) IP 0 0.21 (0.40) 0.16 (0.59) 6 -3.80 (3.14) -0.81 (1.19) 12 -6.70 (4.70) -1.87 (1.54) 24 -9.51 (7.70) -2.16 (1.65) Controls 4 lags of (Z,Y)



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Comparing LP-IV and SVAR-IV - 1 Condition LP-IV⊥

(i) E ε1,t

⊥ Zt⊥′( ) = ′α ≠ 0; (ii)

E ε•,t

⊥ Zt⊥′( ) = 0

(iii) E ε t+ j

⊥ Zt⊥′( ) = 0 for j ≠ 0.

Condition SVAR-IV

(i) E ε1tZt

′( ) = ′α ≠ 0; (ii) E ε•tZt

′( ) = 0

(iii) Invertibility

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Comparing LP-IV and SVAR-IV – 2

Suppose (i) E ε1tZt

′( ) = ′α ≠ 0, (ii) E ε•tZt

′( ) = 0, and

E ε1,t+ jZt

′( ) = 0 for j > 0.

Then Wt = (Yt −1) are valid controls for LP-IV (ensuring

E ε t+ j

⊥ Zt⊥′( ) = 0) if and only if the SMA is invertible.

"No Free Lunch"

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Comparing LP-IV and SVAR-IV – 3

With strong instruments: Regardless of invertibility, Θh,i1

LP− IV is consistent and asy. normal.

With invertibility, Θh,i1

SVAR− IV is more efficient than Θh,i1LP− IV ,

but Θh,i1SVAR− IV is inconsistent without invertibility ("omitted

variable bias"). Hausman-like test based on Θh,i1

SVAR− IV − Θh,i1LP− IV .

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Under strong instruments and "local-to-invertibility"

T θ LP−IV − θSVAR−IV( ) d⎯→⎯ N (d ,V )

ξ= T (θ LP−IV − θSVAR−IV )'V −1(θ LP−IV − θSVAR−IV )

• d is the local departure from invertibility • V = (α2)Ω so instrument strength matters • ξ is a Granger-Causality test (Z → Y), with power

focused on omitted variable bias in SVAR IRFs.

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lag (h) LP-IV SVAR-IV Difference R 0 1.00 (0.00) 1.00 (0.00) 0.00 (0.00) 6 1.12 (0.52) 0.89 (0.31) -0.23 (1.19) 12 0.78 (1.02) 0.78 (0.46) 0.00 (1.79) 24 -0.80 (1.53) 0.40 (0.49) 1.19 (2.57) IP 0 0.21 (0.40) 0.16 (0.59) -0.06 (0.35) 6 -3.80 (3.14) -0.81 (1.19) 3.00 (2.32) 12 -6.70 (4.70) -1.87 (1.54) 4.83 (4.00) 24 -9.51 (7.70) -2.16 (1.65) 7.35 (6.40) Controls 4 lags of (Z,Y)



(Overall tests ξ are also statistically insignificant)

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Left to do

Theory: a little Applications: a lot

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